\section{Introduction}

\subsection{What is Gillies' Conditional?}
\frame{
\hframetitle{What is Gillies' Conditional?}
\vfill
The ``Gillies Conditional'' originates with
\citep{gillies_epistemic_2004}, ``Epistemic conditionals and
conditional Epistemics''
\vfill
Unlike \alert{material implication} or the \alert{counterfactual
  conditional}, Gillies' conditional more directly engages
the Ramsey test and epistemology, trying to model the internal process
one employs when deliberating  on questions like ``If I knew that
$X$, would I know that $Y$''?
\vfill
Forms of this conditional appear informally in
\citep{stalnaker_theory_1968} and \citep{karttunen_syntax_1977}.  The
first formal appearance we are aware of is in
\citep{heim_semantics_1982}.\footnote{We'd like to thank Dr. Katrin Schulz
  for bringing this earlier work on Gillies' epistemic indicative
  to our attention.}
\vfill
However, we shall adhere to Gillies' presentation.  Gillies motivates thinking with a thought experiment\ldots
\vfill
}

\subsection{Motivation}
\begin{frame}[allowframebreaks]

\hframetitle{Motivation}
%To motivate his ideas about conditionalization, 
In \citep{gillies_epistemic_2004} 
we hear the following story:
\begin{quote}
              [Suppose] that there has been a murder in the mansion. The
mansion staff can be partitioned into the grounds staff (people who work outside
the mansion) and the house staff (people who work inside the mansion proper).
Again, the three suspects are the driver (grounds staff), the butler (house staff),
and the gardener (grounds staff). You are the lead investigator, and your young
assistant (who, in fact, is but an apprentice) has been collecting clues and reports
back to you. She has collected this one clue: the butler has an airtight alibi. Your
assistant thus decrees:
\end{quote}
\framebreak
\begin{quote}
\newcounter{gilliesex} % sexual...
\begin{list}{\textup{(\arabic{gilliesex})}}{\usecounter{gilliesex}} 
\item \alert{Therefore: if a member of the grounds staff did it, then
  it was the driver.}
\end{list}

Being a seasoned inspector, you disagree:

\begin{list}{\textup{(\arabic{gilliesex})}}{\usecounter{gilliesex}}
\setcounter{gilliesex}{1}
\item\label{gsex2} \alert{    It's not so that if a member of the grounds
    staff did it, then it was the driver. 
          After all, it might still be the gardener who did it.}
\end{list}

   The truth-functional analysis of conditional statements (i.e., the material conditional) treats an indicative \alert{If $p$, then $q$} as equivalent to the disjunction \alert{not-$p$ or $q$}.
This cannot be right. If the truth-functional analysis were right, then your denial
in \eqref{gsex2} would commit you to accepting that a member of the grounds staff did it
and it was definitely not the driver.
\end{quote}
\end{frame}

\begin{frame}
  \frametitle{Veltman's Mantra}
One way to avoid the above problem is not to commit to a truth functional definition of validity
when thinking about epistemic indicatives.\\
\mvspace{1cm}                                                             
\begin{quote}
Mantra: You know the meaning of a
sentence if you know the change it brings about in the information state of anyone who
accepts the news conveyed by
it. \citep{veltman_defaults_1996}\end{quote}
\mvspace{1cm}
\ldots the conditional that Gillies proposes, based on his thought
experiment uses Veltman style epistemic update semantics.
\end{frame}

\subsection{Syntax \& Semantics}\label{syntaxandsemantics}
\frame[allowframebreaks]
{
  \frametitle{Review of Update Semantics}
To give a formal account of Gillies' conditional, we review (and
slightly adapt) the presentation in 
\citep{van_der_does_updatemight_1997}.

We are going to be concerned with the following grammars, given in \alert{Backus-Naur
form}, where $\Phi$ is a set of proposition letters:
  \begin{itemize}
\item $\mathcal{L}_0$, the language of \alert{propositional logic}:
\[ \phi_0\ {::=}\ p \in \Phi \ |\ \bot \ |\ \neg \phi_0 \ |\ \phi_0 \wedge \psi_0 \]
\item $\mathcal{L}_\Box$, the language of \alert{$\Box$}:
\[ \phi_\Box\ {::=}\ \phi_0 \ | \ \Box \phi_0  \]
\item $\mathcal{L}_\gillies$, the language of \alert{Gillies'
 conditional}:
\[ \phi_\gillies\ {::=}\ \phi_0 \ | \ \phi_0 \gillies \phi_0 \]
  \end{itemize}
\framebreak
Abstract Semantics: Let $\mathbb{A}\ {:=}\  \langle A, \mathbf{1}, \mathbf{0}, \sleq , \sqcap , \sim, \ll\cdot\rr \rangle$ be a \alert{Boolean lattice} with operators $\ll\cdot\rr : A \to \Phi \to A$, which
employ \alert{reverse Polish notation} and conform to the following
axioms:
\mvspace{1cm}
\begin{itemize}
  \item \alert{Reflection}: $\alpha \ll p \rr \sleq \alpha$ for all
    $\alpha \in A$ and $p \in \Phi$
\item \alert{Monotony}: If $\alpha \sleq \beta$ then $\alpha \ll p \rr \sleq \beta \ll p \rr$ for all
    $\{ \alpha, \beta\} \subseteq A$ and $p \in \Phi$
\item \alert{Introspection}: If $\alpha \sleq \beta\ll p \rr $ then $\alpha  = \alpha \ll p \rr$ for all
    $\{ \alpha, \beta\} \subseteq A$ and $p \in \Phi$
%\item \alert{Falsum}: $\alpha[\bot] = \mathbf{0}$ for all $\alpha \in A$
\end{itemize} 
\framebreak
Using the primitive operator $\ll \cdot \rr$, define another
reverse Polish
operator $[\cdot] : A \to \mathcal{L}_0 \cup \mathcal{L}_\gillies
\cup \mathcal{L}_\Box \to A$ recursively as follows, which is referred
to as \alert{epistemic update}:

\begin{itemize}
  \item $\alpha[p]\ {:=}\ \alpha\ll p\rr $
  \item $\alpha[\bot]\ {:=}\ \mathbf{0} $
  \item $\alpha[\neg \phi]\ {:=}\ \alpha \sqcap \sim \alpha[\phi] $
  \item $\alpha[\phi \wedge \psi]\ {:=}\ \alpha[\phi] \sqcap
    \alpha[\psi] $
  \item $\alpha[\Box \psi]\ {:=}\ \begin{cases} \alpha &
      \textup{if } \alpha[\phi]=\alpha\\
        \textbf{0} & \textup{otherwise}\end{cases} $
  \item $\alpha[\phi \gillies \psi]\ {:=}\ \begin{cases} \alpha &
      \textup{if } \alpha[\phi][\psi]=\alpha[\phi]\\
        \textbf{0} & \textup{otherwise}\end{cases} $
\end{itemize}
}

\frame
{
  \frametitle{Intuition}
  \mode<presentation>
  {
    \[ \alpha[\phi \gillies \psi]\ {:=}\ \begin{cases} \alpha &
      \textup{if } \alpha[\phi][\psi]=\alpha[\phi]\\
        \textbf{0} & \textup{otherwise}\end{cases} \]
    }
Why is Gillies' epistemic indicative conditional defined like this?
Gillies offers some intuition in \citep{gillies_epistemic_2004}:

\mvspace{.5cm}
\begin{quote}
I should believe an indicative ``If $p$ then $q$'' just in
case learning $p$ given my present information would be enough to commit me to $q$.
Epistemic conditionals seem to tell us more about the structure of our information
about the world than they do (directly, anyway) about the world.
\end{quote}
}

\frame
{
  \frametitle{Why Must?}
  $\Box$ and Gillies' conditional are intimately linked.

\mvspace{.5cm}

  We found that in doing formal work with the Gillies' conditional, it
  was convenient to elucidate and capitalize on this relationship.

\mvspace{.5cm}

  In the subsequent section, we shall investigate certain key lemmas
  linking $\Box$ and $\gillies$.
}

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